Solutions of functional equations arising in some regression problems and a characterization of the Cauchy law

Abstract

The paper examines in some detail the nature of the probability distribution of the independent and identically distributed random variables (i.i.d.r.v.'s.) X₁,X₂,..., which possess the property E(a₁X₁+a₂X₂+...|b₁X₁+b₂X₂,+...)=0a.s. Both the cases of finite and infinite number of variables are considered. The distribution depends on the nature of the coefficients a₁, b₂, and their relationships.